www.oeaw.ac.at
www.ricam.oeaw.ac.at
Mathematical analysis of the acoustic imaging modality
using bubbles as contrast agents at nearly resonating
frequencies
A. Dabrowski, A. Ghandriche, M. Sini
RICAM-Report 2021-35
MATHEMATICAL ANALYSIS OF THE ACOUSTIC IMAGING MODALITY USING BUBBLES AS CONTRAST AGENTS AT NEARLY RESONATING FREQUENCIES
ALEXANDER DABROWSKI∗, AHCENE GHANDRICHE∗AND MOURAD SINI‡
Abstract. We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves).
We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixedsingle backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region.
(1) We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minnaert resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of these resonances in terms of the background mass density of the background allows us to recover it, i.e.
the mass density, in a straightforward way.
(2) In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well.
1. Introduction and statement of the results
Diffusion by highly contrasted small particles is of fundamental importance in several branches of applied sciences, as for example in material sciences and imaging. In this work, we focus on the acoustic imaging modality using microscaled bubbles as contrast agents, see [10, 17, 20, 21] for more details on related theoretical and experimental studies. We describe a modality using the contrasted scattered fields, by the targeted anomaly, measured before and after injecting microscaled bubbles. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. The main goal of this work is to analyze mathematically this contrasted scattered fields in terms of these scales with incident frequencies close to these resonances and derive explicit formulas linking the values of the unknown mass density and bulk modulus of the targeted region to the measured scattered fields.
To describe properly the mathematical model we are dealing with in this work, let us denote by D a small particle inR3of the formD:=εB+z, whereBis an open, bounded, simply connected set inR3with Lipschitz boundary, containing the origin, andz specifies the location of the particle. The parameterε >0 characterizes
2010Mathematics Subject Classification. 35R30, 35C20.
Key words and phrases. Acoustic imaging, bubbly media, Minnaert resonance, surface-wave resonances, Newtonian potential, body-wave resonances.
∗ RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria. Email: alexan- [email protected]. This author is supported by the Austrian Science Fund (FWF): P 30756-NBL.
∗ RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria. Email:
[email protected]. This author is supported by the Austrian Science Fund (FWF): P 30756-NBL.
‡RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria. Email: [email protected]. This author is partially supported by the Austrian Science Fund (FWF): P 30756-NBL.
1
the smallness assumption on the particle. Let us consider a mass density (respectively, bulk modulus) that we note byρε(·) (respectively,kε(·)) of the form
ρε(x) :=
(ρ0(x), x∈R3\D,
ρ1, x∈D, kε(x) :=
(k0(x), x∈R3\D, k1, x∈D,
whereρ1andk1are positive constants, whileρ0andk0are smooth enough functions which are constant outside of a bounded and smooth domain Ω. We denote respectively ¯ρ0 and ¯k0 to be the values of ρ0 andk0 outside Ω. Thusρ0andk0denote the density and bulk modulus of the background medium, and ρ1 andk1 denote the density and bulk modulus of the bubble respectively.
We are interested in the following problem describing the acoustic scattering by a bubble, see [11] and [12], given by the system
(1.1)
∇ · 1
ρ0
∇u
+ω2 k0
u= 0 inR3\D,
∇ · 1
ρ1
∇u
+ω2 k1
u= 0 inD, u|−−u|+= 0, on∂D,
1 ρ1
∂u
∂ν −
− 1 ρ0
∂u
∂ν +
= 0 on∂D,
where ω > 0 is a given frequency and ν denotes the external unit normal to ∂D. Here the total field is u := ui+us, where ui denotes the incident field (we restrict to plane incident waves) and us denotes the scattered waves which satisfy the following condition
(1.2) ∂us
∂|x|−iκ0us=o 1
|x|
,|x| → ∞, (S.R.C).
We introduce the notation κ20 := ω2ρ0/k0 and κ21 := ω2ρ1/k1. The problem (1.1) is well posed, see [3, 4]
and [9]. In addition, the scattered fielduscan be expanded as us(x, θ) = eiκ0|x|
|x| u∞(ˆx, θ) +O |x|−2
, |x| →+∞,
where ˆx := x/|x| and u∞(ˆx, θ) denotes the far-field pattern corresponding to the unit vectors ˆx, θ, i.e. the incident and propagation directions respectively. We are interested in the regimes where the coefficients satisfy the conditions:
ρ1=Cρεs, s≥0 and k1=Ckεt, t≥0,
with positive constants Cρ and Ck which are independent from ε, and real numbers s, t assumed to be non negative. The scattering problem described above models the acoustic wave diffracted in the presence of small bubbles. In this case, the parameterssandtfix the kind of medium we are considering, see [11, 12, 19] and [9].
Recall that the speed of propagation of the sound isc0 :=q
k0
ρ0 in the background and c1 :=q
k1
ρ1 and in the bubble. We are interested in the following two regimes on the relative speed of propagation inside the bubble as compared to the one in the background, i.e the ratio cc1
0.
(1) Moderate relative speed of propagation. In this case, we assume thats =t, then the relative speed of propagation is uniformly bounded from below and above or
c21
c20 =ρ0k1
k0ρ1
= κ20
κ21 ∼1, asε1.
(2) Small relative speed of propagation. In this case, we assume that s < t, then the relative speed of propagation is small or
c21 c20 =κ20
κ21 ∼εt−s, as ε1.
There is a major difference between these two regimes. In the first one, the contrast between the bubble and the background comes only from the transmission coefficientsρ1/ρ0across the interface of the bubble. Hence, there might be surface waves created by this contrast if it is pronounce enough. For certain scales ofρ1/ρ0, this is indeed the case, as we will see it later. In the second regime, as the speed of propagation is very small inside the bubble, then the sound wave (i.e. the fluctuation) slows down inside it and might create local spots, body waves, even though the bubble has a small size. For certain scales of cc212
0
this is indeed the case.
To highlight these differences, let us for the moment assume thatρ0 is constant everywhere in R3. In this case, the above problem can be equivalently formulated as
∆u+κ20u= 0 inR3\D,
∆u+κ21u= 0 inD, u|ưưu|+= 0, on∂D,
1 ρ1
∂u
∂ν ư
ư 1 ρ0
∂u
∂ν +
= 0 on∂D, uưui satisfies the SRC.
As we can see, the contrasts of the medium appear in the transmission conditions through the coefficient 1/ρ1 (or equivalently ρ0/ρ1), and through the speed of propagation, namely ρ0/k0 and ρ1/k1. Based on the Lippmann-Schwinger equation representation of the total fields, the second contrast appears on the (volumetric) Newtonian potentialNDω defined, fromL2(D) toH2(D), as
NDω[f](x) :=
Z
D
Gω(x, y)f(y)dy,
while the first one appears on the (surface) Neumann-Poincar´e operator (KDω)?defined, fromL2(∂D) toL2(∂D), by
(KDω)?[f](x) :=p.v Z
∂D
∂Gω(x, y)
∂ν(x) f(y)dσ(y).
Precisely, the values of the fielduoutside the bubbleD is fully computable from the knowledge ofu(x), x∈D and∂νu(x), x∈∂D. These last quantities are solutions of the following system of integral equations
(1.3) u(x)ưγω2NDω[u] (x) +αSDω[∂νu] (x) =ui(x), onD and
(1.4) α
1 α+ρ0
2 + (KDω)∗
[∂νu]ưγω2∂νưNDω[u] (x) =∂νui on∂D, whereSDω is the single layer operator defined, fromL2(∂D) toH32(D), as
SDω[f] (x) :=
Z
∂D
Gω(x, y)f(y)dσ(y), andui is the incident field such that
(1.5) div
1 ρ0
∇ui
(x) +ω2 k0
ui(x) = 0, x∈ D.
and we have adapted the succeeding notationsγ=βưαρ1/k1 andα:= 1/ρ1ư1/ρ0 withβ:= 1/k1ư1/k0. Here,Gω stands for the Green’s functions related to (1.5). More precisely,Gω is solution, in the distributional sense, of
(1.6) ∇
x ·h
ρư10 (x)∇
xGω(x, z)i +ω2
k0Gω(x, z) =ưδ
z(x) for anyx, z∈R3. with the radiation conditions at infinity. For ˆxin the unit sphere, we note by
G∞ω(ˆx,·)
the far field associated toGω(·,·). Unless specified, we use the same notationGω(·,·) for the general setting whenρ0andk0 are locally variable.
Depending on the scales of the contrasts, we make the following observations.
(1) In the first regime, i.e s=t, we have γ ∼1, as ε <<1, and the Newtonian potential is negligible as it scales asε2 as ε1. However, ifs=t= 2 then the contrasts on the mass densities, i.e. 1/α, can approximate the spectrum of the Neumann-Poincar´e operatorKD0. For smooth domainD, this operator defined onL2(∂D) has a sequence of real eigenvalues accumulating at 0 in addition to the value−ρ20. As the contrast is real, then we can only approximate the highest eigenvalue, which is−ρ20. This can be done in this regime asα∼ε−2. The frequencyωfor which this is possible is the Minnaert resonance (corresponding to a surface wave type).
(2) In the second regime, if s < t, the high contrasts of the speed of propagation allow the Newtonian operator to dominate the Neumann-Poincar´e operator. In addition, if we take t−s = 2, then the contrast of the speed of propagation,γ∼ε−2, will balance the scale of the Newtonian operator and we might excite its eigenvalues. There is a discrete sequence of such eigenvalues (corresponding to local body waves).
Microbubbles with scales fitting into the first regime are well known to exist in the nature. However, those related to the second regime, with small speed of propagation, are less known. Nevertheless, there are possibilities to artificially produce them, see the discussion in [24] and also in [23].
A first key observation in our analysis, which happens to be useful for the imaging later on, is that the Minnaert resonance and the sequence of body wave resonances are characterized by the bulk modulus of the bubble and the surrounding local mass density of the background. In addition, we show that the contrasted scattered fields reach their maximum values at, incident, frequencies close to the Minneart resonance (or the body-wave resonances depending on the types of bubbles we use). This allows us to recover these resonances by measuring the contrasted scattered waves at a band of incident frequencies but at a fixed single backscattering direction. A second key observation is that this measured contrasted scattered waves allows us to recover the total field at the location point of the bubble. Scanning the targeted region with such bubbles, we can recover the total field there up to a sign (i.e. the total field in the absence of the bubbles).
Based on these observations, we can reconstruct the density and the bulk modulus of the targeted region from the contrasted scattered waves (before and after injecting the bubbles) at a band of incident frequencies but at a fixed single backscattering direction. More details are given in section 2. Nevertheless, let us say it in short here that these contrasted scattered waves encodes the Minnaert and the body-waves resonance in its denominator and the total field in its numerator. From the first one, we extract the mass density while from the second one we derive the bulk modulus of the targeted region.
The following theorems translate these observations with more clear statements. We state the following conditions which are common to both the two results.
Conditions. Let Ωbe a bounded domain of diameterdiam(Ω)of order 1. Let also ρ0 andk0 be two functions of classC1 and are constant outsideΩ. They are assumed to be positive functions. Let D:=z+ε B be a small and Lipschitz smooth domain where z ∈ Ω away from its boundary. The relative diameter of D is small as compared to the diameter ofΩ, i.e. ε
diam(Ω) << 1. The functions ρ0 and k0 are assumed to be independent on the parameterε.
Theorem 1.1. Let the above Conditions be satisfied. In addition, let ρ1 and k1 be constants enjoying the following scales
ρ1=ρ1ε2, k1=k1ε2 and k1
ρ1 ∼1, as ε <<1 andρ1 is large enough such thatmax
x∈Ωρ0(x)< ρ1.
The solution of the corresponding problem (1.1), has the following expansions.
(1) The scattered field is approximated as
(1.7) us(·, θ, ω) =vs(x, θ, ω)− ω2ωM2
k1(ω2−ωM2 )|B|ε Gω(x−z)v(z, θ, ω) +O ε2 (ω2−ωM2 )2
!
uniformly forxin a bounded domain away from D andθin the unit sphere.
(2) The farfield is approximated as
(1.8) u∞(ˆx, θ, ω) =v∞(ˆx, θ, ω)− ρ0ω2ωM2
k1(ω2−ω2M) 4π|B|εv(z,−ˆx, ω)v(z, θ, ω) +O ε2 (ω2−ωM2 )2
! ,
uniformly forθ andxˆ in the unit sphere.
These expansions are valid under the condition that εh
(ω2−ω2M) =O(1), h <1 asε <<1. Here, we have (1.9) ωM :=ωM(z) :=
s 2k1
ρ0(z)µ∂B
, with µ∂B := 1
|∂B|
Z
∂B
Z
∂B
(x−y)·ν(x)
4π|x−y| dσ(x)dσ(y),
called the Minnaert frequency1. In both the expansions v := v(x, θ, ω) = v∞(ˆx, θ, ω) +ui(x, θ) and v∞ :=
v∞(ˆx, θ, ω)is the total field, and is the far field associated to the scattered field, solution of the problem (1.1) in the absence of the bubbleD.
Note, from (1.9), that the Minnaert frequencyωM is such thatωM ∼1, in terms ofε, and sinceωapproaches ωM, we deduce that ω∼1, in terms ofε.
The first mathematical study of the Minnaert resonance was shown in [6] where it was estimated for bubbles injected in a homogeneous background. Later on, a series of works were devoted to its implications in different areas, see [3, 4, 7–9]. The approximations in (1.7) and (1.8) are extensions of those in [6] and [3] to the case when the background is heterogeneous (with variable mass density and bulk modulus). The surprising fact is that this resonance depends also on the surrounding background through its mass density.
To state the results related to the second regime, we first introduce, with some details, the Newtonian operatorNωB defined fromL2(B) to L2(B) as2
NωB[f](x) :=
Z
B
Γω(x, y)f(y)dy= Z
B
ρ0(z) ei ω
qρ
0 (z) k0 |x−y|
4π|x−y| f(y)dy, where Γω(·,·) is the fundamental solution of
∆xΓω(x, z) +ω2ρ0(z)
k0(z) Γω(x, z) =−ρ0(z)δ
z(x), x, z ∈R3,
with radiations conditions at infinity. For ω = 0, the operator N0B[·] is positive, compact, selfadjoint and by ρ0(z)λBn, eBn
n∈Nwe design its sequence of eigenvalues with the corresponding eigenfunctions, where obviously λBn, eBn
n∈Nare related to the operator defined throughf(·)→R
B 1
4π|·−y|f(y)dy.
Theorem 1.2. Let the above Conditions be satisfied. In addition, let ρ1 and k1 be constants enjoying the following scales
(1.10) ρ1=ρ0(z) +O(εj), j >0, and k1=k1ε2 asε <<1.
In this regime, the solution of the problem (1.1), has the following expansions.
1Remark thatµ∂Bdepends only on the shape of the domainB.
2For convenience, we’ll omit the dependency notation ofNωB with respect toz.
(1) The scattered field has the approximation
(1.11) us(x, θ, ω) =vs(x, θ, ω)− 1 k1
ω2ω2n
0
(ω2−ω2n0) Z
B
eBn02
ε Gω(x;z)v(z, θ, ω) +O ε + ε1+min(1;j) ω2−ωn202
! ,
uniformly forxin a bounded domain away from D andθin the unit sphere.
(2) The farfield has the approximation (1.12)
u∞(x, θ, ω) =v∞(x, θ, ω)− ρ0 4π k1
ω2ω2n
0
(ω2−ω2n0) Z
B
eBn
0
2
ε v(z,−ˆx, ω)v(z, θ, ω) +O ε + ε1+min(1;j) ω2−ω2n02
! ,
uniformly forθ andxˆ in the unit sphere.
These expansions are valid as soon as εh ω2−ωn2
0
=O(1), withh <min{1, j}, asε <<1, where
(1.13) ωn0:=
s k1
ρ0(z)λBn
0
.
Observe that R
BeBn02
means P
l
R
BeBl 2
, wherel is such that NB0[eBl ] =λBn0eBl . 3
Here again v := v(x, θ, ω) and v∞ := v∞(ˆx, θ, ω) is the total field, and is the far field associated to the scattered field, solution of the problem (1.1) in the absence of the bubbleD.
The body-wave resonances have been characterized already in [5, 22] in the framework of dielectric nanopar- ticles, in the scalar model related to the TM regime of the electromagnetic scattering, with a homogeneous background. There, the contrast comes from the dielectric nanoparticles with high permittivity and moderate permeability. In our context, the contrast comes from the fact that the density of the bubble is moderate while its bulk is still small. At the mathematical level, our formulas in (1.11) extend those in [5] to the case of the acoustic model, i.e. a divergence form model, with heterogeneous background. As we have said above, such bubble’s contrasts might not be available in nature but can be artificially designed, see [24].
We finish this section with the following observations.
(1) The approximations in Theorem 1.1 are similar to the ones in Theorem 1.2 up to the multiplicative factor appearing in the dominating term. The additional term O(ε) appearing in the error of the approximations (1.11) and (1.12) can be removed as follows:
(a) The scattered fields are approximated as us(x, θ, ω) =vs(x, θ, ω) +ω2
k1 Gω(x, z)v(z, θ, ω) Z
D
W(x)dx+O ε1+min(1;j) ω2−ω2n02
!
(b) The farfields are approximated as u∞(ˆx, θ, ω) =v∞(ˆx, θ, ω) + ρ0ω2
4π k1
v(z,−ˆx, ω)v(z, θ, ω) Z
D
W(x)dx+O ε1+min(1;j) ω2−ω2n
0
2
!
whereW := I−γ ω2N0B
−1
[1]. The termO(ε) appearing in (1.11) and and (1.12), is due to the fact that, see (4.20),
Z
D
W(x)dx=−ωn20 R
DeDn0(x)dx2
ω2−ωn20 +O ε3 .
3This can be seen in (4.20).
(2) In both Theorem 1.1 and Theorem 1.2, the error terms contain the term (ω2−ωε22
M)2 or (ω2−ωε2n20)2, respec- tively. This is not optimal and we believe it can improved to allow us to takeω2−ω2M and ω2−ω2n0 of the order of ε or less. This can be done at the expanse of improving the dominating term of the corresponding approximation. This can be seen from the proofs when the background is homogeneous.
In the inhomogeneous background case, its justification makes the computations rather more involved and we prefer to skip this and stick to the results as stated above for clarity.
(3) Finally, we do believe that the condition max
x∈Ωρ0(x)< ρ1used in Theorem 1.1 and the condition (1.10) appearing in Theorem 1.2 might be removed.
2. An application to the acoustic imaging using resonating bubbles
Based on the expansions given in Theorem 1.1 and Theorem 1.2, in particular (1.8) and (1.12), we design the following imaging procedure to reconstruct the mass density ρ0 and bulk modulus k0 inside the bounded domain Ω where they are variable. This procedure is based on the following measured data.
Let [ωmin, ωmax] be an interval of possible incident frequencies under the following conditions ωmin≤
v u u t
2k1
max
z∈Ωρ0(z)µ∂B ≤ v u u t
2k1
min
z∈Ωρ0(z)µ∂B ≤ωmax.
This condition makes sense as soon as we know a priori a lower bound and an upper bound of the unknown mass densityρ0.
(1) Collect the farfields before injecting the bubble D, i.e. measure the backscattered farfield at a single incident waveθ and a band of frequenciesω∈[ωmin, ωmax] : v∞(−θ, θ, ω).
(2) Collect the farfield after injecting the bubbleD, centered at the pointz∈Ω, i.e. measure the backscat- tered farfield at a single incident waveθand a band of frequenciesω∈[ωmin, ωmax] : u∞(−θ, θ, ω, z).
The imaging procedure goes as follows. We set
(2.1) I(ω, z) :=u∞(−θ, θ, ω, z)−v∞(−θ, θ, ω)
as the imaging functional, remembering that the incident angle θ is fixed. We have the following properties from (1.8)
(2.2) I(ω, z)∼ − ρ0
4π k1
ω2M
(ω2−ωM2 (z))|B|ε [v(z, θ, ω)]2. We divide this procedure into two steps:
(1) Step 1. From this expansion, we recover ω2M(z) as the frequency for which the imaging functionω → I(ω, z) gets its largest value. From the estimation of this resonance ωM2 (z), we reconstruct the mass density at the center of the injected bubblez, based on (1.9), as follows:
(2.3) ρ0(z) = 2k1
ωM2 (z)µ∂B
.
Scanning the domain Ω by such bubbles, we can estimate the mass density there.
(2) Step 2. To estimate now the bulk modulus, we go back to (2.2) or (1.8), and derive the values of the totale field [v(z, θ, ω)]2. This field corresponds to the model without the bubble. Hence, we have at handv(z, θ, ω) forz∈Ω up to a sign (i.e. we know the modulus and the phase up to a multiple ofπ).
Use the equation ∇ ·ρ−10 ∇v+ω2k0−1v = 0 to recover the values of k0 in the regions where v does not change sign. This can be done by numerical differentiation for instance. Other ways are of course possible to achieve this second step. In addition, we have at hand multiple frequency internal data.
The procedure described above uses the Minnaert resonance. The key point to recover the mass density is the explicit dependance of this resonance on the value of the mass density on it’s ’center’, see (1.9). We can do the same work using the sequence of resonances coming from the second regime, see (1.13). Therefore, we may recoverρ0and k0 for this regime as well. However, for technical reasons, we need to know the mass density as we use the condition (1.10). But as we have said earlier, we believe that this condition might be removed.
3. Proof of Theorem 1.1
We divide the proof into two steps. In the first step, we provide the expansions in the case when the background is homogeneous. This allows to show the key parts in localizing the resonance and computing the scattered fields from incident frequencies close to these resonances. In the second step, we deal with the case when the background is heterogeneous and show how this perturbation influences the derivation of the expansions and the resonances as well.
3.1. Constant coefficients. We assume here that bothρ0 andk0 are constants everywhere inR3. We recall thatρ1= ¯ρ1ε2, k1= ¯k1ε2, where ¯ρ1,¯k1 do not depend onε. In this case, it is immediate to show that
Gω(x, y) =ρ0
eiκ0|x−y|
4π|x−y|, x6=y, where κ0:=ωp ρ0/k0. Letube the solution of (1.1). From the Lippman-Schwinger representation we have
(3.1) u(x)−αdiv
x
Z
D
Gω(x, y)∇u(y)dy−β ω2 Z
D
Gω(x, y)u(y)dy=ui(x), where α:= 1/ρ1−1/ρ0 andβ:= 1/k1−1/k0.
Since∇
xGω(x, y) =−∇
yGω(x, y), by integration by parts and (1.6) we have divx
Z
D
Gω(x, y)∇u(y)dy=−ω2ρ1 k1
Z
D
Gω(x, y)u(y)dy− Z
∂D
Gω(x, y)∂νu(y)dσ(y), so (3.1) becomes
(3.2) u(x)−γω2
Z
D
Gω(x, y)u(y)dy+α Z
∂D
Gω(x, y)∂νu(y)dσ(y) =ui(x),
whereγ=β−αρ1/k1.Taking the normal derivative asx→∂Dfrom insideD, from the jump relations of the derivative of the single layer potential we obtain
(3.3)
1 + αρ0 2
∂νu(x)−γω2∂ν−
Z
D
Gω(x, y)u(y)dy+α(KDω)∗[∂νu] (x) =∂νui(x).
Notice that due to the scaling of ρ1 and k1, we have γ = O(1) as ε → 0. Expanding in z the fundamental solution, we obtain forxaway fromD,
Z
D
Gω(x, y)u(y)dy=Gω(x, z) Z
D
u(y)dy+O
ε52kukL2(D)
, as by the Cauchy-Schwartz inequality and the fact that|y−z|=O(ε) we have
Z
D
(y−z)u(y)dy
≤ k · −zkL2(D)kukL2(D)=O
ε52kukL2(D)
. In the same way, we have
Z
∂D
|y−z| ∂νu(y) dσ(y).ε2 k∂νukL2(∂D), so that
Z
∂D
Gω(x, y)∂νu(y)dσ(y) =Gω(x, z) Z
∂D
∂νu(y)dσ(y) +O
ε2k∂νukL2(∂D)
.
Therefore, we can rewrite (3.2) as us(x) =γ ω2Gω(x, z)
Z
D
u(y)dy −α Gω(x, z) Z
∂D
∂νu(y)dσ(y) + O
ε52kukL2(D)+k∂νukL2(∂D)
. From the equation satisfied byu, see for instance (1.1), and the divergence theorem, we have
(3.4)
Z
D
u(y)dy=−k1
ω2 Z
D
∇ · 1
ρ1∇u
(y)dy=− k1
ω2ρ1 Z
∂D
∂νu(y)dσ(y), then
(3.5) us(x) =−
α+γk1 ρ1
Gω(x, z) Z
∂D
∂νu(y)dσ(y) +O
ε52kukL2(D)+k∂νukL2(∂D)
.
Now, we derive the dominating term of R
∂D∂νu dσ and estimate kukL2(D) and k∂νukL2(∂D) in terms of ε.
Let us consider first the case whenγ= 0. In this case, the equation (3.3) becomes (3.6) ((1/α+ρ0/2)I+ (KDω)∗) [∂νu] =α−1∂νui, and we can rewrite it as
(3.7) (1/α+ρ0/2)I+ (KD0)∗
[∂νu] + (KDω)∗−(KD0)∗
[∂νu] =α−1∂νui. Let
(3.8) A∂D:= ρ20
k0
µ∂D, µ∂D:= 1
|∂D|
Z
∂D
Z
∂D
(x−y)·ν(x)
4π|x−y| dσ(x)dσ(y)and A(y) := ρ20 k0
Z
∂D
(x−y)·ν(x) 4π|x−y| dσ(x).
By the divergence theorem we have A∂D >0, and it is immediate thatA−A∂D has average zero along∂D.
ExpandingGω(x, y) in terms of|x−y|, we obtain (KDω)∗[∂νu](x) :=
Z
∂D
∂ Gω(x, y)
∂ν(x) ∂νu(y)dσ(y)
= Z
∂D
ρ0
4π
"
(x−y)·ν(x)
|x−y|3 −1 +iκ0|x−y|
∞
X
n=0
(iκ0|x−y|)n n!
#
∂νu(y)dσ(y)
= (KD0)∗[∂νu](x)−κ20ρ0 8π
Z
∂D
(x−y)·ν(x)
|x−y| ∂νu(y)dσ(y)
− iκ30ρ0 12π
Z
∂D
(x−y)·ν(x)∂νu(y)dσ(y) +O ε3k∂νukL2(∂D)
, (3.9)
and integrating (3.7) on∂D, as KD0[1] =−ρ0/2, see Appendix (5.3), we obtain 1
α−κ20k0
2ρ0
A∂D
Z
∂D
∂νu(x)dσ(x) = 1 α
Z
∂D
∂νui(x)dx+iκ30ρ0
12π Z
∂D
Z
∂D
(x−y)·ν(x)∂νu(y)dydx + κ20k0
2ρ0
Z
∂D
(A(y)−A∂D)∂νu(y)dy+O ε5k∂νukL2(∂D)
. (3.10)
We can estimate the integral which containsA(·)−A∂D by rewriting Z
∂D
(A(y)−A∂D)∂νu(y)dσ(y) (3.6)= α−1 Z
∂D
(A(y)−A∂D) (ρ0/2 + 1/α+ (KDω)∗)−1[∂νui]
(y)dσ(y)
= α−1 Z
∂D
(ρ0/2 + 1/α+KDω)−1[A(·)−A∂D]
(y)∂νui(y)dσ(y)
≤ α−1
(ρ0/2 + 1/α+KDω)−1[A(·)−A∂D] L2(∂D)
∂νui
L2(∂D)=O ε6 , (3.11)
the last equality being a consequence of the fact that (ρ0/2 + 1/α+KDω)−1does not scale on L20(∂D) :={f ∈ L2(∂D) :R
∂Df dσ= 0}, and AandA∂D scale both asε2. Then (3.10) becomes 1
α−iκ30|D|ρ0
4π −κ20k0 2ρ0
A∂D Z
∂D
∂νudσ= 1 α
Z
∂D
∂νuidσ+O ε5k∂νukL2(∂D)+ε6 , where we have used the fact that R
∂D(x−y)·ν(x)dσ(x) = R
Ddiv [x−y]dx = 3|D|. Then, multiplying by α (which scales likeε−2), we obtain the expression of the following dominating term ofR
∂D∂νudσ,
1−i α κ30|D|ρ0
4π −α κ20k0 2ρ0
A∂D Z
∂D
∂νu dσ= Z
∂D
∂νuidσ+O ε3k∂νukL2(∂D)+ε4 .
In the general case ofγ6= 0, instead of identity (3.6), we have 1
α+ρ0
2 + (KDω)∗
[∂νu](x)−ω2γ α ∂ν−
Z
D
Gω(x, y)u(y)dy=α−1∂νui(x).
Integrating on∂D, and integrating by parts the last integral, we obtain Z
∂D
1 α+ρ0
2 +(KDω)∗
[∂νu](x)dσ(x)+ω2γρ0
α ω2
k0
Z
D
Z
D
Gω(x, y)u(y)dy dx+ Z
D
u(x)dx
=α−1 Z
∂D
∂νui(x)dσ(x).
Then, with the same estimates as in (3.10), we obtain 1
α−iκ30|D|ρ0
4π −κ20k0 2ρ0
A∂D Z
∂D
∂νu(x)dσ(x) + ω2γρ0 α
Z
D
u(x)dx=α−1 Z
∂D
∂νui(x)dσ(x)
− ω2γ κ20 α
Z
D
Z
D
Gω(x, y)u(y)dydx+error, (3.12)
where
error:=O ε5k∂νukL2(∂D)+ε6 .
Next, with help of the Cauchy-Schwartz inequality, we estimate the double volume integral as
ω2γ κ20 α
Z
D
Z
D
Gω(x, y)u(y)dydx
.ε112 kukL2(D), then, the equation (3.12) takes the following form
1
α−iκ30|D|ρ0
4π −κ20k0 2ρ0
A∂D
Z
∂D
∂νu(x)dσ(x) +ω2γρ0 α
Z
D
u(x)dx= 1 α
Z
∂D
∂νui(x)dσ(x) +r, where
(3.13) r:=O
ε112kukL2(D)+ε5k∂νukL2(∂D)+ε6 . We use (3.4) and the fact that ∆ui=−κ20ui to obtain
(3.14)
1
α−iκ30|D|ρ0
4π −ω2
2 A∂D−γk1ρ0
αρ1
Z
∂D
∂νu=−ω2ρ0
αk0
Z
D
ui+r=−ω2ρ0
α k0
|D|ui(z) +r.
From the definition ofκ0, recall thatκ0:=ωp
ρ0/k0, we can see that the term between parenthesis on the left hand side of the previous equation is cubic polynomial function onω. Now, we define the Minnaert frequency ωM to be the dominating part of the zero of this cubic polynomial function. To find the dominant part of the zero is equivalent to solve
ω2= 2 A∂D
1
α−iκ30|D|ρ0
4π −γk1ρ0 αρ1
= 2
A∂D
1 α
k1ρ0 ρ1k0
+O(ε) = 2k1 µ∂Bρ0
+O(ε),
where the last two equalities are established using the definition of α, β, γ and the scales of |D| and A∂D. Setting,
(3.15) ω2M := 2k1
µ∂Bρ0
,
we have (3.16)
1
α−iκ30|D|ρ0
4π −ω2
2 A∂D−γk1ρ0 αρ1
=ε2ρ0k1 k0
ωM2 −ω2
ω2M +O ε3 . The equation (3.14), using (3.16), takes the following form
ε2ρ0k1
k0
ω2M−ω2 ω2M
Z
∂D
∂νu(x)dσ(x) +O ε3 Z
∂D
∂νu(x)dσ(x) =−ω2ρ0
α k0 |D|ui(z) +r.
By Cauchy-Schwartz inequality we estimate the second term on the left hand side equation asO
ε4 k∂νuk
L2(∂D)
. Then, we obtain the following formula
Z
∂D
∂νu(x)dσ(x) = κ21 |D| ui(z)
(ω2−ωM2 ) +O ε7+r+ε4 k∂νuk
L2(∂D)
ε2(ω2−ωM2 )
!
(3.13)
= ω2Mκ21 |D|ui(z)
(ω2−ω2M) +O ε4+ε72 kuk
L2(D)+ε2 k∂νuk
L2(∂D)
(ω2−ωM2 )
! (3.17)
= O
ε3 (ω2−ωM2 )
+O ε4+ε72 kukL2(D)+ε2 k∂νukL2(∂D)
(ω2−ωM2 )
! . (3.18)
To estimate the error term, on the last expression, we need the following a priori estimates.
Proposition 3.1. Foru=ui+us, solution of(1.1), it holds
(3.19) k∂νukL2(∂D)=O ε2
ω2−ω2M
, and
(3.20) kukL2(D)=O ε32
ω2−ω2M
! ,
under the condition thatε/ ω2−ωM2
is small enough.
Proof. Let us indicate asC a generic constant independent ofε. From (1.3) we have
(3.21) I−γω2NDω
[u] +α SDω[∂νu] =ui.
Sinceγ =O(1) and thus kNDωkL −−−→ε→0 0, for ε small enough we have thatI−γω2NDω is invertible, so (3.21) takes the following form
u=−α(I−γω2NDω)−1[SDω[∂νu]] + (I−γω2NDω)−1[ui].
Taking theL2(D)-norm in both side of the last equation and using the fact that
I−γω2NDω−1
L(L2(D))≤C to obtain
(3.22) kukL2(D)≤α CkSDω[∂νu]kL2(D)+CkuikL2(D).
In order to finish the last estimation we need to show precisely how the single layer potential scales. For this, by definition, we have
SDω[f]
2
L2(D) :=
Z
D
Z
∂D
Gω(x, y)f(y)dy
2
dx, ∀f ∈L2(∂D)
= ε5 Z
B
Z
∂B
Gεω(η−ξ) ˜f(ξ)dξ
2
dη:=ε5
SBεω [ ˜f]
2 L2(B)
(3.23)
and from the continuity ofSBεω fromL2(∂B) toH32(B) we have that
SDω[f]
2
L2(D)=ε5 SBεω[ ˜f]
2
L2(B)≤ε5C f˜
2
L2(∂B)=ε3C f
2 L2(∂D),
in particular
(3.24)
SDω[∂νu]
L2(D)≤ε32 C
∂νu L2(∂D). Combining (3.22) and (3.24), we obtain
(3.25) kukL2(D)≤ε−12Ck∂νukL2(∂D)+CkuikL2(D).
To manage the termk∂νukL2(∂D)we use the boundary integral equation given by (1.4), to write (3.26) ∂νu=α−1
1 α+ρ0
2 + (KDω)∗ −1
∂νui +ω2γ
α 1
α+ρ0
2 + (KDω)∗ −1
[∂νNDω[u]] on ∂D.
Next, for convenience, we set
T :=
1 α+ρ0
2 + (KDω)∗ −1
and we rewrite (3.26) as
∂u
∂ν = 1 α T
∂ui
∂ν − 1
|∂D|
Z
∂D
∂ui
∂ν
+ 1
|∂D|
Z
∂D
∂ui
∂ν 1 α T [1]
+ ω2γ α T
∂NDω[u]
∂ν− − 1
|∂D|
Z
∂D
∂NDω[u]
∂ν−
+ω2γ α
1
|∂D|
Z
∂D
∂NDω[u]
∂ν− T [1]. (3.27)
Since −ρ20 is an eigenvalue of KD0 with associated eigenspace consisting of constant functions, we have the estimates
(3.28) kTkL(L2(∂D)) =
1 α+ρ0
2 + (KDω)∗ −1
L(L2(∂D))
≤Cα,
and on the space of functions with zero average we have
(3.29) kTkL(L2
0(∂D))=
1 α+ρ0
2 + (KDω)∗ −1
L(L2
0(∂D))
≤C.
Now, take theL2(∂D)-norm in both side of (3.27), with the help of (3.28) and (3.29) we obtain
∂u
∂ν
L2(∂D) . α−1
∂ui
∂ν − 1
|∂D|
Z
∂D
∂ui
∂ν L2
0(∂D)
+ 1
|∂D|
Z
∂D
∂ui
∂ν
k1kL2(∂D)
+ α−1
∂NDω[u]
∂ν− − 1
|∂D|
Z
∂D
∂NDω[u]
∂ν− L20(∂D)
+ 1
|∂D|
Z
∂D
∂NDω[u]
∂ν−
k1kL2(∂D). (3.30)
Obviously, we have (3.31)
Z
∂D
∂ui
∂ν
= Z
D
∆ui
=|κ20| Z
D
ui
=O ε3 and, by the triangular inequality and the smoothness of∂νui, we obtain (3.32)
∂ui
∂ν − 1
|∂D|
Z
∂D
∂ui
∂ν L2
0(∂D)
.
∂ui
∂ν L2
0(∂D)
=O(ε). We also have, recalling the definition of the Green function,
Z
∂D
∂NDω[u]
∂ ν (x)dx = −ρ0
Z
D
u(x)dx−ω2ρ0
k0 Z
D
Z
D
Gω(x, y)u(y)dydx
= −ρ0
Z
D
u(x)dx+O
ε72 kukL2(D)
(3.4)
= k1ρ0
ω2ρ1 Z
∂D
∂νu(x)dσ(x) +O
ε72 kukL2(D)
